impact angle control guidance synthesis for evasive

Copyright ⓒ The Korean Society for Aeronautical & Space SciencesReceived: May 12,2017 Revised: September 7, 2017 Accepted: September 11, 2017

719 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

PaperInt’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719

Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile

Y. H. Yogaswara*Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

Department of Research and Development of Indonesian Air Force, Bandung 40174, Indonesia

Seong-Min Hong** and Min-Jea Tahk*** Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

Hyo-Sang Shin**** Cranfield University, Bedford MK43 0AL, United Kingdom

Abstract

This paper proposes a synthesis of new guidance law to generate an evasive maneuver against enemy’s missile interception

while considering its impact angle, acceleration, and field-of-view constraints. The first component of the synthesis is a new

function of repulsive Artificial Potential Field to generate the evasive maneuver as a real-time dynamic obstacle avoidance.

The terminal impact angle and terminal acceleration constraints compliance are based on Time-to-Go Polynomial Guidance

as the second component. The last component is the Logarithmic Barrier Function to satisfy the field-of-view limitation

constraint by compensating the excessive total acceleration command. These three components are synthesized into a new

guidance law, which involves three design parameter gains. Parameter study and numerical simulations are delivered to

demonstrate the performance of the proposed repulsive function and guidance law. Finally, the guidance law simulations

effectively achieve the zero terminal miss distance, while satisfying an evasive maneuver against intercept missile, considering

impact angle, acceleration, and field-of-view limitation constraints simultaneously.

Key words: Missile guidance, Evasive maneuver, Impact angle control, Artificial potential field

1. Introduction

Since an Integrated Air Defense Systems (IADS) [1] have

been sophistically developed, a countermeasure action to

counteract the IADS becomes a significant consideration in

a missile design. Surface attack missiles, which are launched

from air or surface platforms to attack designated surface

targets also need advanced solutions to respond the threats

of IADS. Guidance system design for surface attack missile in

this high threat environment is challenging since the attacking

missile must be delivered to its target, while maximizing

the survivability from the IADS. A proper guidance laws to

generate an evasive maneuver against intercept missiles

are rarely published in open literature. On the contrary,

the evasive maneuver of manned aircraft against intercept

missile has been studied extensively. Those evasive strategies

are based on continuously changing maneuver direction

such as barrel roll and the Vertical-S maneuver [2, p. 120] or

weaving maneuver [2, Ch. 27]. The strategies are not adequate

to be implemented into homing missiles or unmanned aerial

vehicles (UAVs) due to several reasons, i.e.: the movement can

be easily predicted, the maneuver might exceed seeker’s field-

of-view (FOV) limitation, and the missile fails to satisfy zero

terminal miss distance.

This is an Open Access article distributed under the terms of the Creative Com-mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

* Ph. D Student, Research Officer ** Ph. D Student *** Professor, Corresponding author: [email protected] **** Associate Professor

(719~728)2017-97.indd 719 2018-01-06 오후 7:13:25

DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719 720

Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)

An optimal evasive maneuver for subsonic Anti-Ship

Missiles (ASM) has been studied against Close-in Weapon

System (CIWS) [3]. The CIWS is a common short-range IADS

for naval ships that are usually equipped with cannon as

the defensive weapon. Since the ASM moves with constant

acceleration at any moment, the CIWS is able to aim at the

predicted intercept position. In [3], the authors have defined

a time-varying weighted sum of the inverse of the aiming

errors as a cost function which to be minimized. Inspired

by the study above, a direct optimization technique using

Co-Evolutionary Augmented Lagrangian Method (CEALM)

was applied in [4], which the capturability analysis is proven

using Lyapunov-like approach. Both studies in [3] and [4]

generate a type of barrel roll evasive maneuver, however, a

large barrel roll maneuver could increase the miss distance

at the terminal time.

As studied in [5], the typical CIWS’s cannon system has

limitations due to single-threat-engagement capability and

fire at a predicted intercept point of the threat’s trajectory.

This limitation motivates a new guidance law by using salvo

attack where multiple missiles are launched and guided to

arrive at a stationary target at the same desired time. The

guidance law adopting this idea has been firstly proposed by

Jeon et al. in [6] and called as Impact Time Control Guidance

(ITCG). To control the impact time, the ITCG suggests an

additional loop in a closed form solution based on the

linear formulation of traditional proportional navigation

guidance (PNG) law. The ITCG has been also proposed in

[7] by introducing a virtual leader approach. Sliding mode

control (SMC) method for ITCG has been adopted in [8]. A

Lyapunov-based law is the latest method proposed in [9] for

ITCG and simultaneous arrival.

Even though the ITCG has been broadly studied, the

concept of salvo attack for simultaneous impact time by

exploiting the limitation of CIWS is less effective. In fact,

if we may be allowed for a little exaggeration, it is not

applicable anymore. This knowledge is exposed since

CIWS or IADS introduces intercept missiles to enhance the

previous cannon system. The intercept missile has advanced

capability of multi-target engagement, longer intercept

range, and proximity warhead detonation for a higher

probability of kill. Regarding those new developments,

this paper proposes a new guidance law, which maximizes

survivability of an attack missile with respect to the threat of

advanced CIWS or IADS. In order to neutralize the threat of

intercept missile of enemy’s IADS, our attack missile must

have an evasive maneuver capability to counteract the

interception. In addition, the attack missile must satisfy an

impact angle constraint and zero terminal acceleration for

maximizing warhead detonation effect on the target. FOV

limitation of the attack missile is also considered to ensure

the seeker locks on the target.

Due to its elegant concept and simple computation, the

evasive maneuver of the attack missile in this paper is based

on the concept of Artificial Potential Field (APF). The APF

was introduced by Krogh [10] and Khatib [11] for mobile

robots by defining functions of goal attractive potentials and

obstacle repulsive potentials. Inherent problems of APF were

systematically identified by Koren and Borenstein in [12]

due to a trap situations caused by local minima, no passage

between close space obstacle, and oscillation. In addition,

Goal Non-reachable with Obstacle Nearby (GNRON)

problem was also recognized and solved by Ge and Cui in

[13]. Modifications of APF for moving obstacles have been

also proposed in [14]–[16]. Nevertheless, those modifications

cannot be implemented to the missile problem since the

modified potential functions require a deceleration variable

to reduce velocity when facing the obstacle. Eventually,

inspired by Chen et al. in [17], this paper proposes a new

repulsive function that effective for the missile problem

and overcomes the GNRON problem at the same time.

The primary limitation of APF on trap situation due to

local minima will not be an issue in this evasive maneuver

scenario since the problem is one-on-one engagement with

relatively free dead-end trajectory, e.g. U-shaped obstacle.

Regarding the concept of APF, instead of the conventional

attractive potential function, this paper implements an

impact angle control guidance (IACG) method as the

attractive goal. The IACG attacks the weak spot of a target

to increase warhead detonation effect and maximize

its probability of kill towards the target. A generalized

formulation of energy minimization had been proposed in

[18] to achieve IACG. A PNG-based was proposed in [19]

for capturing all possible terminal impact angle. Lyapunov-

based pursuit guidance was introduced in [20] to reduce

the angle between the velocity and the distance vector. A

combination of differential game and sliding mode control

were also proposed in [21]. The Time-to-go Polynomial

Guidance (TPG) as proposed by Lee et al. in [22] which

treat the guidance command as a function of time-to-go is

the most suitable to be adopted in this paper. In addition to

the impact angle constraint, the TPG also satisfies the zero

terminal acceleration to minimize the terminal angle-of-

attack (AOA) for effective impact angle, and zero terminal

lateral velocity to minimize zero effort miss.

In order to improve the practicality on the physical

concern of proposed guidance law, the limitation on

actuator’s acceleration command is considered in this paper.

Moreover, the seeker look angle is also limited to be inside the

FOV in order to ensure the missile seeker not losing its target.

(719~728)2017-97.indd 720 2018-01-06 오후 7:13:25

721

Y. H. Yogaswara Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile

http://ijass.org

Switching logic was implemented in [23] by switching the

guidance law from ITCG of [6] to the second law of constant

look angle guidance command, when the seeker look angle

exceeds the FOV limit. A rule of the cosine of a weighted

leading angle in the biased term was used in [24] to ensure

the ITCG was satisfied without violating the FOV constraint

during an engagement. Switching logic to constant seeker

look angle was also investigated in [25] for optimal IACG

of a missile with a strap-down seeker. Without applying the

switching logic, SMC was used in [26] to satisfy IACG by

implementing a control Integral Barrier Lyapunov Function

(iBLF) to design the reaching law. The implementation of

iBLF inspires this paper to apply a simpler internal penalty

function known as Logarithmic Barrier Function (LBF) to

limit the look angle.

This paper is organized as follows. Section 2 presents the

new proposed repulsive potential functions of APF with

its properties. In Section 3, the equations of motion, APF,

TPG, LBF and all corresponding problem formulation are

synthesized to generate a single acceleration command of

proposed guidance law. Guidance characteristics, analysis,

and solutions are presented in Section 4 by numerical

simulation at a particular missile engagement scenario. The

simulations successfully demonstrate the effectiveness of

the potential function and guidance law. Finally, concluding

remarks and future works are given in Section 5.

2. Proposed Repulsive Potential Function

The repulsive potential function that is implemented on

mobile robots, robot manipulators, and UAVs are broadly

based on the Khatib [11] or Ge and Cui [13]. However,

those repulsive potential functions are not suitable to be

used in missile engagement problem due to its potential

characteristics. The main properties of those functions is

a steep slope of repulsive potential, which the potential

values ascent sharply approaching the obstacle position.

This property is acceptable for mobile robots since it has the

capability of deceleration when approaching the obstacle,

stop the movement, and reroute its path. Conversely, the

missiles engagement scenario does not recognize those

capabilities.

The repulsive potential function for missile engagement

scenario needs a gradual ascent of potential approaching the

obstacle to anticipate the intercept in advance. The gradual

and gentle ascent of repulsive potential has been introduced

for UAV path planning by Chen et al. in [17]. Unfortunately,

this function does not consider the GNRON problem, which

drives a failure to achieve the goal position. The failure is

generated by shifting out the global minimum from the goal

position when the goal is within the influence distance of the

obstacle.

A new repulsive potential function of APF is constructed

and proposed to generate a gradual ascent of potential

approaching the obstacle and solve the GNRON problem

at the same time. Assuming a generic problem of a point

masses vehicle, goal, and obstacle, which the vehicle moves

in a two-dimensional (2-D) space. The vehicle position in

the workspace is denoted by

5

problem formulation are synthesized to generate a single acceleration command of proposed guidance law.

Guidance characteristics, analysis, and solutions are presented in Section 4 by numerical simulation at a

particular missile engagement scenario. The simulations successfully demonstrate the effectiveness of the

potential function and guidance law. Finally, concluding remarks and future works are given in Section 5.


The repulsive potential function that is implemented on mobile robots, robot manipulators, and UAVs

are broadly based on the Khatib [11] or Ge and Cui [13]. However, those repulsive potential functions are

not suitable to be used in missile engagement problem due to its potential characteristics. The main

properties of those functions is a steep slope of repulsive potential, which the potential values ascent sharply

approaching the obstacle position. This property is acceptable for mobile robots since it has the capability of

deceleration when approaching the obstacle, stop the movement, and reroute its path. Conversely, the

missiles engagement scenario does not recognize those capabilities.

The repulsive potential function for missile engagement scenario needs a gradual ascent of potential

approaching the obstacle to anticipate the intercept in advance. The gradual and gentle ascent of repulsive

potential has been introduced for UAV path planning by Chen et al. in [17]. Unfortunately, this function does

not consider the GNRON problem, which drives a failure to achieve the goal position. The failure is

generated by shifting out the global minimum from the goal position when the goal is within the influence

distance of the obstacle.

A new repulsive potential function of APF is constructed and proposed to generate a gradual ascent of

potential approaching the obstacle and solve the GNRON problem at the same time. Assuming a generic

problem of a point masses vehicle, goal, and obstacle, which the vehicle moves in a two-dimensional (2-D)

space. The vehicle position in the workspace is denoted by [ ]TA Ax y=q . The repulsive potential ( )repU q

at each vehicle position is defined by considering the relative distance of goal position into Chen’s equation

as follow

goal , if( )

0, if

obstdobst o

repobst o

e d d dU

d d

ζe − ≤= >

q (1)

. The repulsive

potential

5
























as follow

goal , if( )

0, if

obstdobst o

repobst o

e d d dU

d d

ζe − ≤= >

q (1)

at each vehicle position is defined by

considering the relative distance of goal position into Chen’s

equation as follow

5
























as follow

goal , if( )

0, if

obstdobst o

repobst o

e d d dU

d d

ζe − ≤= >

q (1),

(1)

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)

where obstd , goald , 0d , e , and ζ are the minimal distance between the vehicle and the obstacle, the

distance between the vehicle and the goal, the distance of influence of the obstacle, and both are positive

design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero

as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.

The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-

dimensional (1-D) space as shown in Fig. 1. The vehicle [ ]0 TAx=q is moving along x-axis toward the

goal [ ]4 0 Tgoal =q while avoiding the obstacle [ ]0 0 T

obst =q . Assuming the distance of influence of the

obstacle 0 6d = , the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is

demonstrated in the first plot series. Since the goal position near the obstacle, the generated repulsive

potential is large enough to create the non-reachable goal. This problem takes place since the goal position is

affected by the obstacle and drive non-zero potential at the goal. Moreover, the potentials are evenly

distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the

new proposed function shows significant improvement to handle the GNRON problem. The plot of three

different combinations of scaling gains maintains the minimum of the potential at the goal position and the

maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to

control the properties of repulsive potential. The higher value of e, the higher peak value of the potential. The

higher value of ζ, the steeper ascent of potential approaching the obstacle.

-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep

e=3, ζ=0.4 (Chen)e=2, ζ=0.4e=1, ζ=0.4e=1, ζ=0.3ObstacleGoal

Fig. 1. Repulsive potential function in a 1-D spaceThe corresponding repulsive force is given by the negative gradient of the repulsive potential. According

to Eq.(1), when the vehicle is not at the goal, i.e., goal≠q q , the repulsive force is given by

,(2)

where dobst, dgoal, d0, ε, and ζ are the minimal distance

between the vehicle and the obstacle, the distance between

the vehicle and the goal, the distance of influence of the

obstacle, and both are positive design parameter gains,

respectively. This proposed function ensures the repulsive

potential approaches zero as the vehicle approaches the goal

and finally the goal position will be the global minimum of

total potential.

The effectiveness of the proposed repulsive potential

function is demonstrated in a case on one-dimensional

(1-D) space as shown in Fig. 1. The vehicle

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)



















-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep




is

moving along x-axis toward the goal

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)



















-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep




while

avoiding the obstacle

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)



















-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep




. Assuming the distance

of influence of the obstacle d0=6, the GNRON problem of the

predecessor function as mentioned by Chen et al. in [17] is

demonstrated in the first plot series. Since the goal position

near the obstacle, the generated repulsive potential is large

enough to create the non-reachable goal. This problem takes

place since the goal position is affected by the obstacle and

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)



















-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep




Fig. 1. Repulsive potential function in a 1-D space

(719~728)2017-97.indd 721 2018-01-06 오후 7:13:26



drive non-zero potential at the goal. Moreover, the potentials

are evenly distributed to the right and the left side of the

obstacle neglecting the goal. In the same case assumption,

the new proposed function shows significant improvement

to handle the GNRON problem. The plot of three different

combinations of scaling gains maintains the minimum of

the potential at the goal position and the maximum of the

potential at the obstacle position. Furthermore, the scaling

gains show the freedom to control the properties of repulsive

potential. The higher value of ε, the higher peak value of

the potential. The higher value of ζ, the steeper ascent of

potential approaching the obstacle.

The corresponding repulsive force is given by the negative

gradient of the repulsive potential. According to Eq.(1), when

the vehicle is not at the goal, i.e.,

6

goal goal

obst obst

d

d

= −

= −

q q

q q(2)



















-6 -4 -2 0 2 4 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position, x

Rep

ulsi

ve P

oten

tial,

U rep



to Eq.(1), when the vehicle is not at the goal, i.e., goal≠q q , the repulsive force is given by the repulsive force

is given by

7

( )( ) ( )

, if( )

0, if

rep rep

repObst obst repGoal goal obst orep

obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)

obstdrepObst goalF d e ζeζ −= (4)

obstdrepGoalF e ζe −= (5)

where obst obstd= ∇n and goal goald= −∇n are unit vectors pointing from the obstacle to the vehicle and from

the vehicle to the goal, respectively. Those unit vectors play an important role since the obstn repulses the

vehicle away from the obstacle and the goaln attracts the vehicle towards the goal.

To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the

scaling gains are defined as 20e = and 0.3ζ = . The repulsive potential field and repulsive force field of

the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal

as the global minima and the potential peak at the obstacle. The repulsive potential force represents the

potential as a positive divergent vector field outward the obstacle and a negative divergent vector field

inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and

attracted to the goal.

-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y

Repulsive ForceObstacleGoal

Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space

Regarding the implementation of the new repulsive function into missile evasive maneuver, some

nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,

the intercept missile, and the target, respectively.

3. Guidance Synthesis

Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The

,

(3)

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








, (4)

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








, (5)

where

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








and

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








are unit vectors

pointing from the obstacle to the vehicle and from the

vehicle to the goal, respectively. Those unit vectors play an

important role since the

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








repulses the vehicle away from

the obstacle and the

7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y








attracts the vehicle towards the

goal.

To elaborate the properties of the force field, the case on

Fig. 1 is developed into 2-D space, which the scaling gains

are defined as ε=20 and ζ=0.3. The repulsive potential field

and repulsive force field of the vehicle at every position in

a 2-D space are depicted in Fig. 2. The repulsive potential

field keeps the goal as the global minima and the potential

peak at the obstacle. The repulsive potential force represents

the potential as a positive divergent vector field outward the

obstacle and a negative divergent vector field inward the

goal. Intuitively, the vehicle that affected by this vector field

will be repulsed by the obstacle and attracted to the goal.

Regarding the implementation of the new repulsive

function into missile evasive maneuver, some nomenclatures

are adjusted. The vehicle of interest, the obstacle and the

goal are defined as the attack missile, the intercept missile,

and the target, respectively.


Consider a two-dimensional homing guidance scenario

as shown in the left illustration of Fig. 3. The friendly attack

missile has a constant velocity VA heading to enemy’s

stationary target while avoiding enemy’s intercept missile,

1

1. Affiliation’s postcode (우편번호) of 1st author:

Department of Research and Development of Indonesian Air Force, Bandung 40174, Indonesia

2. Affiliation’s postcode (우편번호) of 4th author:

Cranfield University, Bedford, MK43 0AL, United Kingdom

3. Change right figure of Fig. 3 (simplified vector):

Fig. 3. Guidance geometry (left) and guidance synthesis of acceleration command vector (right)


7

( )( ) ( )

, if( )

0, if

rep rep


obst o

F U

F F d dF

d d

= −∇

+ ≤= >

q q

n nq

(3)













-6 -4 -2 0 2 4 6 -10

0

10

0

10

20

30

40

50

60

70

80

90

YX

Rep

ulsi

ve P

oten

tial,

U rep

ObstacleGoal

-4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

4

X

Y









(719~728)2017-97.indd 722 2018-01-06 오후 7:13:34

723


http://ijass.org

which has a constant velocity VI. Acceleration command a

is perpendicular to velocity vector to change the flight path

angle γ of each missile. The position of the attack missile, the

intercept missile, and the target are denoted as (xA, yA), (xI, yI),

and (xT, yT), respectively. Their relationships are denoted as

follows; relative range R(.), relative velocity V(.), line-of-sight

(LOS) angle σ(.), and seeker look angle λ(.). The subscripts 0,

f, A, I, TA, IA denote the initial time, terminal time, attack,

intercept, relationship of the attack missile regarding the

target, and the attack missile regarding the interceptor,

respectively.

The equation of motion in this homing problem for both

attack and intercept missile in inertial frame are generally

given by

8

friendly attack missile has a constant velocity AV heading to enemy’s stationary target while avoiding

enemy’s intercept missile, which has a constant velocity IV . Acceleration command a is perpendicular to

velocity vector to change the flight path angle γ of each missile. The position of the attack missile, the

intercept missile, and the target are denoted as ( ),A Ax y , ( ),I Ix y , and ( ),T Tx y , respectively. Their

relationships are denoted as follows; relative range ( )R ⋅ , relative velocity ( )V ⋅ , line-of-sight (LOS) angle ( )σ ⋅ ,

and seeker look angle ( )λ ⋅ . The subscripts 0, , , , ,f A I TA IA denote the initial time, terminal time, attack,

intercept, relationship of the attack missile regarding the target, and the attack missile regarding the

interceptor, respectively.


The equation of motion in this homing problem for both attack and intercept missile in inertial frame are

generally given by

cos

sin

dx Vdtdy Vdtd adt V

γ

γ

γ

=

=

=

(6)

With its boundary conditions are defined as follows:

( ) ( ) ( )( ) ( ) ( )

0 0 0 0 0 0

f f f f f f

x t x y t y t

x t x y t y t

γ γ

γ γ

= = =

= = =

The guidance law is derived by using small angle approximation of AOA, that the velocity vector and body

orientation nearly have the same value. Hence, the seeker look angle of attack missile toward the target can

.

(6)


8

friendly attack missile has a constant velocity AV heading to enemy’s stationary target while avoiding

enemy’s intercept missile, which has a constant velocity IV . Acceleration command a is perpendicular to

velocity vector to change the flight path angle γ of each missile. The position of the attack missile, the

intercept missile, and the target are denoted as ( ),A Ax y , ( ),I Ix y , and ( ),T Tx y , respectively. Their

relationships are denoted as follows; relative range ( )R ⋅ , relative velocity ( )V ⋅ , line-of-sight (LOS) angle ( )σ ⋅ ,

and seeker look angle ( )λ ⋅ . The subscripts 0, , , , ,f A I TA IA denote the initial time, terminal time, attack,

intercept, relationship of the attack missile regarding the target, and the attack missile regarding the

interceptor, respectively.


The equation of motion in this homing problem for both attack and intercept missile in inertial frame are

generally given by

cos

sin

dx Vdtdy Vdtd adt V

γ

γ

γ

=

=

=

(6)


( ) ( ) ( )( ) ( ) ( )

0 0 0 0 0 0

f f f f f f

x t x y t y t

x t x y t y t

γ γ

γ γ

= = =

= = =

The guidance law is derived by using small angle approximation of AOA, that the velocity vector and body

orientation nearly have the same value. Hence, the seeker look angle of attack missile toward the target can

.

The guidance law is derived by using small angle

approximation of AOA, that the velocity vector and body

orientation nearly have the same value. Hence, the seeker

look angle of attack missile toward the target can be

approximated as subtraction of flight path and LOS angle

9

be approximated as subtraction of flight path and LOS angle

λ γ σ= − (7)

Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law

A SYNa a= , which a synthesis of three components is simply

LBFSYN TPG AFPa a a a= + + (8)

This formulation is described as follow. Principally, the guidance synthesis implements the APF concept by

defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than

implementing classical attractive force function of APF, the acceleration command of TPG TPGa is

preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal

acceleration. The new proposed repulsive force function of APF performs as the second component to

generate the evasive maneuver. Recalling Eq. (3), the acceleration command of APF is taken from the

proposed repulsive potential force ( )APF repa F= q . Once the intercept missile approaching, the repulsive

force is generated as APFa and producing a new resultant vector of ( )TPG AFPa a+ avoiding the interceptor.

Finally, acceleration command of LBF LBFa is also generated when the seeker look angle close to its FOV

limit by compensating the exceeding acceleration command. Through this model, the synthesized guidance

law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss

distance, terminal impact angle, zero terminal acceleration, and FOV limitation.

3.1 Time-to-Go Polynomial Guidance

The TPG demonstrates an effective IACG not only its ability to satisfies the terminal impact angle, but

also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero

terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration

command TPGa and the estimation of time-to-go got for the curved path of the attack missile can be

formulated as follows:

( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go

Va t m n t m n t m nt

σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

. (7)

Recalling the right illustration of Fig. 3, the total

acceleration command of the guidance law

9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

,

which a synthesis of three components is simply

9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

. (8)

This formulation is described as follow. Principally,

the guidance synthesis implements the APF concept

by defining the attractive potential to be achieved and

repulsive potential to be avoided. Rather than implementing

classical attractive force function of APF, the acceleration

command of TPG aTPG is preferred to achieve zero terminal

miss distance at designated terminal impact angle and

zero terminal acceleration. The new proposed repulsive

force function of APF performs as the second component

to generate the evasive maneuver. Recalling Eq. (3), the

acceleration command of APF is taken from the proposed

repulsive potential force aAPF=Frep(q). Once the intercept

missile approaching, the repulsive force is generated as

aAPF and producing a new resultant vector of (aTPG+aAFP)

avoiding the interceptor. Finally, acceleration command of

LBF aLBF is also generated when the seeker look angle close

to its FOV limit by compensating the exceeding acceleration

command. Through this model, the synthesized guidance

law propose a responsive approach to achieve an effective

evasive maneuver while satisfying zero miss distance,

terminal impact angle, zero terminal acceleration, and FOV

limitation.


The TPG demonstrates an effective IACG not only its

ability to satisfies the terminal impact angle, but also satisfies

the zero terminal acceleration to minimize the terminal AOA

for precise impact angle, and zero terminal lateral velocity

to minimize zero effort miss. Recalling the TPG on [22], the

missile acceleration command aTPG and the estimation of

time-to-go tgo for the curved path of the attack missile can be


9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

,

(9)

9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10)

9


λ γ σ= − (7)






















( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 3 1 1ATPG A f

go


σ γ γ = − − + + + + + + + + (9)

( ) ( )( ) ( ) ( ){ }2 2 21 11 2 32 21go A f TA f A f TA f

A

Rt P P PV

γ γ σ γ γ γ σ γ = + − − − + − − − (10),

(10)

10

( )( )( )( )( )

1

2

3

12 3 2 3 3

2 2

3 32 2

Pm n m n

P m n

P m n

=+ + + +

= + +

= + +

(11)

where m and n denote the guidance gains which are chosen to be any positive real values following

0n m> > for zero terminal acceleration. If 1n = and 0m = , the performance of applied TPG results are

identical to the optimal guidance laws with terminal impact constraints, but without zero terminal

acceleration as studied in [18]. Higher values of m and n gains will not only satisfy desired impact angle but

also produce the zero terminal acceleration to avoid saturating commands and sufficiently small terminal

AOA to increase the lethality of the warhead.

On the other hand, the enemy’s intercept missile applies a Pure Proportional Navigation (PPN) in order

to intercept the attack missile. The PPN is chosen due to its natural characteristics in a practical sense as

concluded by Shukla and Mahapatra in [27]. Referring to the literature, the acceleration command for the

intercept missile is formulated as

I I IA

AI AIIA

AI AI

a NVR VR R

σ

σ

= −×

=

�

(12)

where the navigation constant is defined as N = 3.

3.2 Logarithmic Barrier Function

In a real application, the target should be located inside the FOV of the attack missile, and it is important

to keep the seeker look angle from exceeding the limitation. Regarding the look angle on conventional PNG

that decreases to zero as the missile approaches its target, the proposed guidance law is intended to achieve

an additional capability. This capability generates an uncommon trajectory that increases the look angle up to

exceeds the FOV. When the missile fails to lock on the target, it leads the missile into a huge miss distance

and unsatisfied constraints at the terminal phase. Introducing the FOV limit as the barrier b and the barrier

parameter µ , the final component in Eq. (8) can be easily derived by using LBF. Implementing the

characteristics of LBF into a compensated acceleration command ensures the command increases to

actuator’s limitation maxa as the current look angle approaching the barrier and keep the seeker look angle

,

(11)

where m and n denote the guidance gains which are chosen

to be any positive real values following n>m>0 for zero

terminal acceleration. If n=1 and m=0, the performance of

applied TPG results are identical to the optimal guidance

laws with terminal impact constraints, but without zero

terminal acceleration as studied in [18]. Higher values of m

and n gains will not only satisfy desired impact angle but also

produce the zero terminal acceleration to avoid saturating

commands and sufficiently small terminal AOA to increase

the lethality of the warhead.

On the other hand, the enemy’s intercept missile applies

a Pure Proportional Navigation (PPN) in order to intercept

the attack missile. The PPN is chosen due to its natural

characteristics in a practical sense as concluded by Shukla

and Mahapatra in [27]. Referring to the literature, the

acceleration command for the intercept missile is formulated

as

(719~728)2017-97.indd 723 2018-01-06 오후 7:13:35



2

4. Image file of equation 12 and 13:

5. Change right figure of Fig. 5 (change plotted variable):

Fig. 5. Seeker look angle (left), flight path angle (center), and acceleration command (right)

0 10 20 30 40 50 60

-40

-20

0

20

40

60

Time t, sec

Look

Ang

le

, deg

Seeker Look Angle (o = f = -30o )

Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

FOV = lim= 40o

0 10 20 30 40 50 60 70-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le

, deg

Flight Path Angle (o = f = -30o )

Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

o = f = -30o

0 10 20 30 40 50 60-80

-60

-40

-20

0

20

40

60

80

100

X: 58.14Y: 0.06908

Time, sec

a com

, m/s

2

Acceleration Command of Guidance Synthesis ( =90, =3e-4, =10 )

aTPG

aAPFaLBF

aSYN

alim = 30 m/s2

,

(12)

where the navigation constant is defined as N = 3.

3.2 Logarithmic Barrier Function

In a real application, the target should be located inside

the FOV of the attack missile, and it is important to keep the

seeker look angle from exceeding the limitation. Regarding

the look angle on conventional PNG that decreases to zero

as the missile approaches its target, the proposed guidance

law is intended to achieve an additional capability. This

capability generates an uncommon trajectory that increases

the look angle up to exceeds the FOV. When the missile fails

to lock on the target, it leads the missile into a huge miss

distance and unsatisfied constraints at the terminal phase.

Introducing the FOV limit as the barrier b and the barrier

parameter μ, the final component in Eq. (8) can be easily

derived by using LBF. Implementing the characteristics of

LBF into a compensated acceleration command ensures the

command increases to actuator’s limitation alim as the current

look angle approaching the barrier and keep the seeker look

angle inside the FOV in a simple way. Acceleration command

in component of LBF can be formulated as

2




0 10 20 30 40 50 60

-40

-20

0

20

40

60

Time t, sec

Look

Ang

le

, deg


Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

FOV = lim= 40o

0 10 20 30 40 50 60 70-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le

, deg


Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

o = f = -30o

0 10 20 30 40 50 60-80

-60

-40

-20

0

20

40

60

80

100

X: 58.14Y: 0.06908

Time, sec

a com

, m/s

2


aTPG

aAPFaLBF

aSYN

alim = 30 m/s2

.

(13)

Finally, by substituting Eqs. (3), (9), and (13) into (8), the

total acceleration command of attack missile is synthesized

as a new proposed guidance law.

3.3 Survivability of Attack Missile

The survivability or Probability of Survival Ps of the

attack missile is important to be quantified to measure the

effectiveness of the proposed repulsive potential function and

guidance law. As introduced in [28, Ch. 1], the survivability

is the complement of the killability or Probability of Kill Pk

which quantifies the probability of the aircraft being killed.

Thus, the relationship can be formulated as

11

inside the FOV in a simple way. Acceleration command in component of LBF can be formulated as

( ) ( )( )

max

o

o

if

log if 5

0 if 5

LBF

a b

a b b b

b

λ

µ λ λ

λ

≥= − − > ≥ −

< −

. (13)

Finally, by substituting Eqs. (3), (9), and (13) into (8), the total acceleration command of attack missile is

synthesized as a new proposed guidance law.

3.3 Survivability of Attack Missile

The survivability or Probability of Survival SP of the attack missile is important to be quantified to

measure the effectiveness of the proposed repulsive potential function and guidance law. As introduced in

[28, Ch. 1], the survivability is the complement of the killability or Probability of Kill KP which quantifies

the probability of the aircraft being killed. Thus, the relationship can be formulated as below.

1 KSP P= − (14)

From a practical point of view, the killability KP is a complex function, which is explained in detail in the

reference [29, Ch. 6.3]. The intercept missiles typically use a fragmentation warhead, which at a particular

distance to its target, detonates the explosive charge and breaks up the warhead case into smaller accelerating

fragments. The killability for this fragmentation warhead is defined as a function of five parameters, i.e.: the

presented target area, the vulnerable target area, the total number of fragments, spray angle of fragments, and

miss distance at detonation. Recalling the reference, to achieve 0.9KP = , the number of hits on a target

with a vulnerable area is minimum 10% of the presented area, which needs a minimum of 22 hits from the

total fragmentation at the target.

Since all parameters in the reference are given for a rocket baseline against a drone aircraft, those

assumptions can be partially adopted into this paper. The first assumption is that all the five parameters are

known to achieve 0.9KP = . Secondly, the function of the killability will only depend on miss distance at

detonation detd . Finally, following the typical relation trends of KP and detd , a simple relation is

introduced as

. (14)

From a practical point of view, the killability Pk is a

complex function, which is explained in detail in the

reference [29, Ch. 6.3]. The intercept missiles typically use

a fragmentation warhead, which at a particular distance

to its target, detonates the explosive charge and breaks up

the warhead case into smaller accelerating fragments. The

killability for this fragmentation warhead is defined as a

function of five parameters, i.e.: the presented target area,

the vulnerable target area, the total number of fragments,

spray angle of fragments, and miss distance at detonation.

Recalling the reference, to achieve Pk=0.9, the number of hits

on a target with a vulnerable area is minimum 10% of the

presented area, which needs a minimum of 22 hits from the

total fragmentation at the target.

Since all parameters in the reference are given for a rocket

baseline against a drone aircraft, those assumptions can

be partially adopted into this paper. The first assumption

is that all the five parameters are known to achieve Pk=0.9.

Secondly, the function of the killability will only depend on

miss distance at detonation ddet. Finally, following the typical

relation trends of Pk and ddet, a simple relation is introduced

as

12

0

2

detexpKd

dPR

= −

(15)

where0dR is defined as the reference distance. In order to achieve the 0.9KP = at det 6md = ,

018.5 mdR = is chosen to fit the curve.

4. Numerical Analysis

4.1 Performance of Proposed Guidance Law

Let us consider a typical Suppression of Enemy Air Defenses (SEAD) mission of our attack missile with

constant velocity. The attack missile is in its terminal phase maintaining the constant velocity and terminal

impact angle towards the stationary target. The defending enemy’s intercept missile is planted in front of the

target facing the attack missile. The intercept missile will be launched at a range where the attack missile can

be intercepted and neutralized in constant and 10% faster velocity than the attack missile. The intercept

missile has a designated time of flight due to propellant burn time limitation and, if the time exceeds, it is

considered that the intercept missile fails to neutralize the attack missile. To implement the proposed

guidance law, the position of intercept missile and target are assumed to be clearly detected by the attack

missile as a result of active radar or passive seeker detection. A parameter study is carried out to elaborate the

performance of evasive maneuver due to design parameters gains [ ], ,e ζ µ . The quantities of each

parameter for the SEAD engagement scenario are listed in Table 1.

,

(15)

where

12

0

2

detexpKd

dPR

= −

(15)
















is defined as the reference distance. In order to

achieve the Pk=0.9 at ddet=6m,

12

0

2

detexpKd

dPR

= −

(15)
















=18.5m is chosen to fit the

curve.



Let us consider a typical Suppression of Enemy Air

Defenses (SEAD) mission of our attack missile with

constant velocity. The attack missile is in its terminal phase

maintaining the constant velocity and terminal impact

angle towards the stationary target. The defending enemy’s

intercept missile is planted in front of the target facing the

attack missile. The intercept missile will be launched at

a range where the attack missile can be intercepted and

neutralized in constant and 10% faster velocity than the

attack missile. The intercept missile has a designated time

of flight due to propellant burn time limitation and, if the

time exceeds, it is considered that the intercept missile fails

to neutralize the attack missile. To implement the proposed

guidance law, the position of intercept missile and target

are assumed to be clearly detected by the attack missile

as a result of active radar or passive seeker detection. A

parameter study is carried out to elaborate the performance

of evasive maneuver due to design parameters gains [ε, ζ, μ].

The quantities of each parameter for the SEAD engagement

scenario are listed in Table 1.

(719~728)2017-97.indd 724 2018-01-06 오후 7:13:36

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As seen in Fig. 4, the performance of evasive maneuver

is highly sensitive to the gains [ε, ζ, μ]. By defining LBF gain

μ=0, the performances of APF gains [ε, ζ] are evaluated on

the left and center of the Fig. 4. Recalling the properties of

the repulsive potential field, the steepness of the potential is

verified by an early evasive maneuver when the attack missile

enters the distance of influence radius d0. With respect to

the non-evasive maneuver, the evasive deviation width is

determined by the value of the APF gains [ε, ζ]. Starting from

ε≥70 with constant ζ, the attack missile is able to escape the

interception and gives wider evasive trajectories for higher

values. At a constant ε=90, wider evasive trajectories are also

demonstrated for higher values of ζ. Wide evasive trajectory

maximizes the escape of the attack missile, but intuitively

generates a huge flight path angle and exceeding FOV. For

the values of ζ ≥3e-4, the guidance law needs a compensation

command to suppress the flight path angle to ensure its

seeker look angle inside FOV limit. Right graph of Fig. 4

illustrates the effect for different values of LBF gain μ which

higher value of μ suppresses the guidance of attack missile

for narrower path trajectory. Regarding this parameter study,

gain combinations of ε=[90,100], ζ=3e-4, μ=[5,10] are picked for

further analysis since it complies with the initial requirement

to achieve zero miss distance and escaping an interception.

Elaboration of look angle λ suppression, flight path angle

γ, and acceleration command aSYN constraints are time

series plotted in Fig. 5. Once the look angle approaching

the barrier value λlim=±40o, the LBF gains μ=[5,10] generate a

suppression command to keep the look angle inside the FOV.

The higher value of gain, the more suppression applied to

the look angle. However, gain values μ≤2 in this scenario are

not adequate to maintain the look angle from violating the

limit. From the parametric study, the value μ=10 is a rational

choice to satisfy both constraints and will be used for further

examination. Depressed look angle when approaching

FOV limit affects the flight path angle correspondingly. By

implementing the IACG of TPG, the impact angle constraint

is satisfied even though its basic guidance law is combined

with other components by guidance synthesis. All impact

angle histories are relaxed into impact angle constraint γ0=-

14

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

2

3

4

5

6

7

8

Crossrange, km

Dow

nran

ge, k

m

Trajectory of Evasive Maneuver (Parameter Study of Gain e )

Non-Evasive

e= 70, ζ=1e-4, µ=0

e= 90, ζ=1e-4, µ=0

e=110, ζ=1e-4, µ=0Intercept MissileIntercept PlatformStationary Target

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

2

3

4

5

6

7

8

Crossrange, km

Dow

nran

ge, k

m

Trajectory of Evasive Maneuver (Parameter Study of Gain ζ )

Non-Evasive

e=90, ζ=1e-4, µ=0

e=90, ζ=2e-4, µ=0

e=90, ζ=3e-4, µ=0


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

2

3

4

5

6

7

8

Crossrange, km

Dow

nran

ge, k

m

Trajectory of Evasive Maneuver (Parameter Study of Gain µ )

Non-Evasive

e=90, ζ=3e-4, µ=2

e=90, ζ=3e-4, µ=5

e=90, ζ=3e-4, µ=10


Fig. 4. Parameter study of APF gain e (left), APF gain ζ (center), and LBF gain µ (right)

Elaboration of look angle λ suppression, flight path angle γ, and acceleration command SYNa constraints

are time series plotted in Fig. 5. Once the look angle approaching the barrier value olim 40λ = ± , the LBF

gains [ ]5, 10µ = generate a suppression command to keep the look angle inside the FOV. The higher value

of gain, the more suppression applied to the look angle. However, gain values 2µ ≤ in this scenario are

not adequate to maintain the look angle from violating the limit. From the parametric study, the value

10µ = is a rational choice to satisfy both constraints and will be used for further examination. Depressed

look angle when approaching FOV limit affects the flight path angle correspondingly. By implementing the

IACG of TPG, the impact angle constraint is satisfied even though its basic guidance law is combined with

other components by guidance synthesis. All impact angle histories are relaxed into impact angle constraint

o30oγ = − , and the look angle converges to zero. Recalling Fig. 5, components of acceleration command

express the synthesis of the new guidance laws for the gain setting490, 3 , 10ee ζ µ−= = = . The synthesized

total attack command SYNa effectively satisfies the terminal zero acceleration of TPG and keeps inside

actuator’s acceleration limit 2max 30a ms−= ± .

0 10 20 30 40 50 60

-40

-20

0

20

40

60

Time t, sec

Look

Ang

le λ

, deg

Seeker Look Angle (γo = γf = -30o )

Non-Evasive

e=90, ζ=3e-4, µ= 0

e=90, ζ=3e-4, µ= 5

e=90, ζ=3e-4, µ=10

FOV = λlim= ±40o

0 10 20 30 40 50 60 70-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le γ

, deg

Flight Path Angle (γo = γf = -30o )

Non-Evasive

e=90, ζ=3e-4, µ= 0

e=90, ζ=3e-4, µ= 5

e=90, ζ=3e-4, µ=10

γo = γf = -30o

0 10 20 30 40 50 60-80

-60

-40

-20

0

20

40

Time, sec

a com

, m/s

2

Acceleration Command of Guidance Synthesis ( e=90, ζ=3e-4, µ=10 )

aTPG

aAPFaLBF

aSYN

amax = ±30 m/s2


Fig. 4. Parameter study of APF gain ε (left), APF gain ζ (center), and LBF gain μ (right)

2




0 10 20 30 40 50 60

-40

-20

0

20

40

60

Time t, sec

Look

Ang

le

, deg


Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

FOV = lim= 40o

0 10 20 30 40 50 60 70-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le

, deg


Non-Evasive

=90, =3e-4, = 0

=90, =3e-4, = 5

=90, =3e-4, =10

o = f = -30o

0 10 20 30 40 50 60-80

-60

-40

-20

0

20

40

60

80

100

X: 58.14Y: 0.06908

Time, sec

a com

, m/s

2


aTPG

aAPFaLBF

aSYN

alim = 30 m/s2


Table 1. Parameters of SEAD engagement scenario

13

Table 1. Parameters of SEAD engagement scenario

Parameter Symbol Value

Target, stationary position ( ),T Tx y (4, 0) km

Attack missile, initial position ( )0 0,A Ax y (-5, 6) km

Intercept missile, initial position ( )0 0,I Ix y (0, 0) km

Attack missile, velocity AV 200 m/s

Intercept missile, velocity IV 220 m/s

Attack missile, initial flight path angle 0γ -30 deg

Attack missile, terminal impact angle fγ -30 deg

Intercept missile, time of flight - 20 s

Gain set of TPG ( ),m n (2, 3)

Distance of Influence of Obstacle 0d 6 km

As seen in Fig. 4, the performance of evasive maneuver is highly sensitive to the gains [ ], ,e ζ µ . By

defining LBF gain 0µ = , the performances of APF gains [ ],e ζ are evaluated on the left and center of the

Fig. 4. Recalling the properties of the repulsive potential field, the steepness of the potential is verified by an

early evasive maneuver when the attack missile enters the distance of influence radius 0d . With respect to

the non-evasive maneuver, the evasive deviation width is determined by the value of the APF gains [ ],e ζ .

Starting from 70e ≥ with constant ζ, the attack missile is able to escape the interception and gives wider

evasive trajectories for higher values. At a constant 90e = , wider evasive trajectories are also demonstrated

for higher values of ζ. Wide evasive trajectory maximizes the escape of the attack missile, but intuitively

generates a huge flight path angle and exceeding FOV. For the values of 43eζ −≥ , the guidance law needs a

compensation command to suppress the flight path angle to ensure its seeker look angle inside FOV limit.

Right graph of Fig. 4 illustrates the effect for different values of APF gain µ which higher value of µ

suppresses the guidance of attack missile for narrower path trajectory. Regarding this parameter study, gain

combinations of [ ] [ ]490, 100 , 3 , 5, 10ee ζ µ−= = = are picked for further analysis since it complies with the

initial requirement to achieve zero miss distance and escaping an interception.

(719~728)2017-97.indd 725 2018-01-06 오후 7:13:37



30o, and the look angle converges to zero. Recalling Fig. 5,

components of acceleration command express the synthesis

of the new guidance laws for the gain setting ε=90, ζ=3e-4,

μ=10. The synthesized total attack command aSYN effectively

satisfies the terminal zero acceleration of TPG and keeps

inside actuator’s acceleration limit alim=±30ms-2.

4.2 Interception Survivability

Applying the same engagement scenario as studied

above, the attack missile is simulated at its terminal phase

maintaining the prescribed impact angle, constant velocity,

and launched at all possible region above target. The initial

flight path angle of attack missile is determined as the

initial LOS angle to the target and defined as the terminal

impact angle to be achieved(γf =γ0=σ0). Fig. 6 compares the

survivability map of the attack missile with evasive maneuver

regarding the survivability map without evasive maneuver.

The higher survivability, towards Ps=1, represents that

the attack missile has more probability to survive from

interception. In contrast, the lower survivability towards

Ps=0 represents the attack missile has more probability to

be intercepted and neutralized before hitting the target. By

only implementing a single set of gain ε=90, ζ=3e-4, μ=10,

the guidance law demonstrates its effectiveness to increase

the survivability of the attack missile on the SEAD mission

scenario. The proposed guidance law extends the area of

initial position for the attack missile, which has the high

survivability to accomplish the mission.

4.3 Implementation in Operational Scenario

Based on the parametric study and considering the

survivability map, the trajectory plot in Fig. 7 elaborates

three operational scenarios of broader initial position

inside the initial distance of influence. Those scenarios

represent a practical implementation of the guidance law

based on a single gain set ε=90, ζ=3e-4, μ=10. Different initial

positions, i.e.: [-4.9, 3.7] km, [-1.5, 4.8] km, and [1.5, 5.2] km

are defined as Scenario 1, 2, and 3 respectively to represent

a general engagement of SEAD scenario. In each scenario,

three trajectories of simulation result are displayed, i.e.:

the trajectory of attack missile with and without evasive

maneuvers, and the trajectory of the intercept missile.

3

6. Change both figure of Fig. 6 (high resolution):

Fig. 6. Survivability map as a function of initial position

Fig. 6. Survivability map as a function of initial position without evasive maneuver (left) and with evasive maneuver (right)

16

scenario, three trajectories of simulation result are displayed, i.e.: the trajectory of attack missile with and

without evasive maneuvers, and the trajectory of the intercept missile. Since the engagement scenario

implementing the proposed guidance law, the trajectories clearly validate the effectiveness of the guidance

law to escape from interception by generating an evasive maneuver for different and wider range of initial

position.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

2

3

4

5

6

7

8

Crossrange, km

Dow

nran

ge, k

m

Trajectory of Evasive Maneuver against Intercept Missile (e=90, ζ=3e-4, µ=10)

Scenario 1Scenario 2Scenario 3Intercept PlatformStationary Target

Fig. 7. Trajectory of operational scenario

Furthermore, as seen in Fig. 8, the constraints of terminal impact angle, look angle limitation, and zero

terminal acceleration are all satisfied in the three operational scenarios. Since the impact angles are defined

as the initial flight path ( )0 fγ γ= respectively, the terminal impact angles are all smoothly achieved in each

scenario. The look angles are also suppressed to keep inside the limitation of FOV and converge to zero look

angle. Finally, in addition to limit the acceleration command, the terminal accelerations are also converged to

zero. All results verify that the synthesis of new guidance law based on the new proposed repulsive potential

function is effective and reliable to be implemented as a new solution for evasive maneuver against intercept

missiles.

0 5 10 15 20 25 30 35 40 45 50 55-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le γ

, deg

Flight Path Angle (γo = γf) - Operational Scenario

Scenario 1Scenario 2Scenario 3

0 5 10 15 20 25 30 35 40 45 50 55-50

-40

-30

-20

-10

0

10

20

30

40

50

Time t, sec

Look

Ang

le λ

, deg

Seeker Look Angle (γo = γf) - Operational Scenario


FOV = λlim= ±40o

0 5 10 15 20 25 30 35 40 45 50 55-40

-30

-20

-10

0

10

20

30

40

Time, sec

a com

, m/s

2

Acceleration Command of Guidance Synthesis - Operational Scenario


alim = ±30 m/s2

Fig. 8. Flight path angle (left), look angle (center), and acceleration command (right)of operational scenariosFig. 8. Flight path angle (left), look angle (center), and acceleration command (right) of operational scenarios

16

scenario, three trajectories of simulation result are displayed, i.e.: the trajectory of attack missile with and

without evasive maneuvers, and the trajectory of the intercept missile. Since the engagement scenario

implementing the proposed guidance law, the trajectories clearly validate the effectiveness of the guidance

law to escape from interception by generating an evasive maneuver for different and wider range of initial

position.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

2

3

4

5

6

7

8

Crossrange, km

Dow

nran

ge, k

m

Trajectory of Evasive Maneuver against Intercept Missile (e=90, ζ=3e-4, µ=10)

Scenario 1Scenario 2Scenario 3Intercept PlatformStationary Target


Furthermore, as seen in Fig. 8, the constraints of terminal impact angle, look angle limitation, and zero

terminal acceleration are all satisfied in the three operational scenarios. Since the impact angles are defined

as the initial flight path ( )0 fγ γ= respectively, the terminal impact angles are all smoothly achieved in each

scenario. The look angles are also suppressed to keep inside the limitation of FOV and converge to zero look

angle. Finally, in addition to limit the acceleration command, the terminal accelerations are also converged to

zero. All results verify that the synthesis of new guidance law based on the new proposed repulsive potential

function is effective and reliable to be implemented as a new solution for evasive maneuver against intercept

missiles.

0 5 10 15 20 25 30 35 40 45 50 55-100

-80

-60

-40

-20

0

20

Time t, sec

Flig

ht P

ath

Ang

le γ

, deg

Flight Path Angle (γo = γf) - Operational Scenario


0 5 10 15 20 25 30 35 40 45 50 55-50

-40

-30

-20

-10

0

10

20

30

40

50

Time t, sec

Look

Ang

le λ

, deg

Seeker Look Angle (γo = γf) - Operational Scenario


FOV = λlim= ±40o

0 5 10 15 20 25 30 35 40 45 50 55-40

-30

-20

-10

0

10

20

30

40

Time, sec

a com

, m/s

2

Acceleration Command of Guidance Synthesis - Operational Scenario


alim = ±30 m/s2

Fig. 8. Flight path angle (left), look angle (center), and acceleration command (right)of operational scenarios


(719~728)2017-97.indd 726 2018-01-06 오후 7:13:38

727


http://ijass.org

Since the engagement scenario implementing the

proposed guidance law, the trajectories clearly validate the

effectiveness of the guidance law to escape from interception

by generating an evasive maneuver for different and wider

range of initial position.

Furthermore, as seen in Fig. 8, the constraints of terminal

impact angle, look angle limitation, and zero terminal

acceleration are all satisfied in the three operational

scenarios. Since the impact angles are defined as the initial

flight path (γ0=γf) respectively, the terminal impact angles

are all smoothly achieved in each scenario. The look angles

are also suppressed to keep inside the limitation of FOV and

converge to zero look angle. Finally, in addition to limit the

acceleration command, the terminal accelerations are also

converged to zero. All results verify that the synthesis of new

guidance law based on the new proposed repulsive potential

function is effective and reliable to be implemented as a new

solution for evasive maneuver against intercept missiles.

5. Conclusion

This paper proposes a synthesis of new guidance law

to generate an evasive maneuver against enemy’s missile

interception while considering its impact angle, acceleration,

and FOV constraints. The guidance law introduces a simple

approach but an effective result for a real-time avoidance

against dynamic obstacles. The new guidance law synthesizes

three components, starting with the new repulsive potential

function of APF to generate the evasive maneuver. The

zero terminal miss distance is satisfied by TPG as well as to

satisfy the impact angle and zero terminal acceleration. The

guidance law is finally synthesized with the LBF to guarantee

the look angle inside the FOV. The parametric study on the

gains of [ε, ζ, μ] are carried to generate a reliable gain set.

SEAD engagement scenarios of attack missile headed for

a stationary target that is defended by an intercept missile

are performed on numerical simulation. The gain set ε=90,

ζ=3e-4, μ=10 demonstrates the expected performance of

guidance law. The commanded acceleration is proven to

generate the evasive maneuver of attack missile avoiding

the enemy’s missile interception. The avoidance trajectory

of attack missile is adequate to escape the interception

while managing the constraints of impact angle, zero impact

acceleration, and FOV limit. Without the evasive maneuver,

the attack missile cannot survive the interception. However,

the survivability of the attack missiles is enhanced when the

guidance law is applied. The enhancement of survivability

is clearly compared in survivability maps for the scenarios

with and without evasive maneuver. Briefly, the proposed

guidance law and new repulsive potential function perform

a simple, reliable and effective approach to generate evasive

maneuver against missile intercept while satisfying the

constraints.

Study on the optimization of gain values can be carried out

as further works. Optimized scaling gains will improve the

guidance performance and also enhance the survivability in a

broader area of initial position. An expansion of probability map

using optimized gains will ensure the success of engagement

mission over a high threat environment. Implementation of

the proposed guidance law for multiple missiles and another

engagement geometry such as air-to-air or surface-to-air

scenarios are also considered for future works.

Acknowledgement

Y. H. Yogaswara expresses his appreciation to Indonesia

Endowment Fund for Education (LPDP RI), for providing

scholarship on a doctoral program at the Korea Advanced

Institute of Science and Technology, Republic of Korea.

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